The extended affine Weyl group

Why do translations and length-zero rotations describe the same indexing group?

1 · Start with matrices

· Split linear model Definition

Let $\resfield$ be an algebraically closed field, $n\ge1$, and $\unif$ a uniformizer.

$$\OO=\resfield[\![\unif]\!].$$ $$\FF=\resfield(\!(\unif)\!).$$ $$G=\GLn.$$ $$T=\{\text{invertible diagonal matrices}\}.$$ $$B=\{\text{invertible upper triangular matrices}\}.$$ $$\Iwa=\{g\in\GLn(\OO):g\bmod\unif\in B\}.$$

The same matrix formulas hold over $\mathbb F_q$. We use the algebraically closed field for the geometric affine flag variety.

2 · The Concept

· The extended affine Weyl group Definition

With $N_G(T)$ the normalizer of $T$,

$$\Wt:=N_G(T)(\FF)/T(\OO)\simeq\ZZ^n\rtimes S_n.$$

It records a diagonal valuation vector and a permutation; it indexes the Iwahori double cosets.

$$\Iwa\backslash\GLn(\FF)/\Iwa\simeq\Wt.$$

[HR, Def. 7; Prop. 8]

3 · Follow the lattice chain

Define the rotation by its action on column vectors:

$$\rot:(v_1,\ldots,v_n)\longmapsto(v_2,\ldots,v_n,\unif v_1),\qquad\LLk:=\rot^k\OO^n\quad(k\in\ZZ).$$

4 · Reading path

Origin of the problem → two splittings → GL₁, GL₂, GL₃ → proof of the lattice formula → transfer tasks.

Afterward: multiply monomial representatives; distinguish $\Wa$ from $\Wt$; compute $\rot^k\OO^n$; explain why a nontrivial translation can have length zero.

5 · Mathematical status

Definitions are labelled; the GLₙ calculations are proved here. The general double-coset and splitting results are cited precisely in Source.

Return to IV · Affine space.